Interference is the phenomenon whereby the wave vectors (amplitude and phase) of two or more waves (i.e. electromagnetic waves, sound waves etc.) are combined as they propagate along a common path. An interferometer is a device that generates this interference between two (or more) waves. The waves are first separated into two (or more) paths in order to measure a quantity that differentiates these paths (i.e. spatial or material differences). A phase difference between the waves in each path results from these differences. When the waves are recombined, the interference pattern can be used to measure this phase difference and provide information about the spatial and material differences between the paths. One of these paths is called the reference path and the other(s) is(are) the test path(s). The reference path is usually one that has well known spatial and material properties such as free space (vacuum) or air. The problem with interferometers today is that they all suffer from issues related the production and use of this physical reference arm.
Interferometers require one test arm (at least) and a reference arm. The reference arm is a physical path with well known characteristics. The most common reference arm is one that contains a variable free space path (such as a variable delay line—VDL) that can be used to change the length of the reference arm and/or balance the two arms of the interferometer.
Several possible implementations of physically balanced interferometers are given in FIG. 1. The examples shown are the Michelson Interferometer, the Single Arm Interferometer and the Mach Zehnder Interferometer. Fiber based (optical) implementations are shown but a similar diagram can be made for free space architectures or architectures that do not involve optical waves (i.e. sound waves). Note that the source can be any wave generating source such as, for example an RF source, microwave source, laser source, broadband source, X-ray source, sound wave generator etc. The detector/receiver can be any wave receiver such as for example an RF receiver, microwave receiver, laser detector, broadband receiver (optical spectrum analyser, monochromater or even a basic tuneable filter), X-ray detector, sound wave receiver. The discussion that follows assumes the use of a tunable laser as the source and a laser detector as the receiver. The choice of this pair for the source and detector is only to simplify the discussion and illustrate tangibly the issues related to using a physical reference and are not meant to limit the generality of the discussion.
An interferometer is balanced when the delay experienced by the wave traversing the reference arm is the same as the delay experienced by the wave traversing the test arm. When an interferometer is brought into balance the resulting interference pattern depends on the type of interferometer and the number of interfering waves. Various types of interferometers are shown in FIG. 1 and will be used in the discussion to follow.
Note that the following discussions involve the use of spectral interferometry which uses a tunable wavelength source and a detector to produce interference patterns with the wavelength as the dependent variable (can also be done spectrally using a broadband source and an optical spectrum analyzer—or any device that can discriminate between power levels at different wavelengths). This is not meant to limit the generality of this technology to spectrally acquired interference patterns. Interference patterns can also be produced by exploiting the temporal or spatial coherence of two waves. This can be done, for example, by using a broadband source, detector and a movable mirror. In this case, instead of using a laser that tunes (moves) its wavelength a mirror is moved so that the interference pattern produced is a function of the position of the movable mirror (located in the test path of an unbalanced interferometer). This type of Interferometry is temporal Interferometry. An application of this type of Interferometry is Fourier Transform Spectroscopy.
Dual Arm Interferometer
Several types of interferometers exist. The most common are interferometers with two (dual) arms. Some examples of dual arm interferometers are the Michelson Interferometer and the Mach Zehnder Interferometer (shown in FIG. 1). In a dual arm interferometer there are two interfering waves (U0 and U1). Let us take the Michelson interferometer as an example. The interfering waves in the Michelson interferometer can be described in the following way:U0=Ae(−2jβL1) U1=Ae(−2jkoL2) 
It has been assumed, for simplicity in this example, that the amplitudes of the reflected waves are equal (given by ‘A’) and that the simulated reference path is free space (given by the propagation constant k0). Additionally, in this example, a general propagation constant, β, has been assumed for the test path. The interference pattern produced by the interference of U0 and U1 can be described as the square of the magnitude of the two interacting waves:I=[U0+U1]2=A2(2+2 cos(2(βL1−koL2)))
This interference pattern is shown in FIG. 2(a). From this we can see that the phase of the interference pattern is related to the difference between the two paths.φ=2(βL1−koL2)
The phase, therefore, can be used to obtain information about the test arm (described by βL1) based on the knowledge of the reference arm (described by koL2). Note that ko is the free space propagation constant (ko=2π/λ) where λ is the wavelength. L1 is the length of the test arm and L2 is the length of the reference arm.
Single Arm Interferometer
In a Single Arm Interferometer there are three interfering waves (U0, U1 and U2). Let us assume for simplicity we will choose to make that the magnitude of the three interfering waves (‘A’) equal. The three interfering waves of a Single Arm Interferometer can be described in the following way:U0=A U1=Ae(−2jβL1) U2=Ae(−2jβL1−2jkoL2) 
The interference pattern produced by the interference of U0, U1 and U2 can be described as the square of the magnitude of the three interacting waves:I=[U0+U1+U2]2=A2(3+2 cos(2(βL1+koL2))+4 cos(βL1+koL2)cos(βL1−koL2))
This interference pattern is shown in FIG. 2(b). From this we can see that the phase of the interference pattern is related to the difference between the two paths.φ=(βL1−koL2)
Once again the phase can be used to obtain information about the test arm (described by βL1) based on the knowledge of the reference arm (described by koL2). Note that ko is the free space propagation constant (ko=2π/λ) where λ is the wavelength. L1 is the length of the test arm and L2 is the length of the reference arm.
From the preceding discussion one can see that the interference patterns obtained by dual and single arm interferometers are equivalent since the phase of the envelope (top half of the amplitude modulation) in the case of the single arm interferometer and the phase of the actual interference pattern in the case of the dual arm interferometer is related to a difference between the two paths. FIG. 2(a) illustrates the interference patterns generated by dual arm (2 wave) interferometers and FIG. 2(b) illustrates the interference pattern generated by Single arm (3 wave) interferometers. FIG. 2 illustrates how the phase of the slowly varying ‘envelope’ in a Single Arm Interferometer is equivalent to the phase of the two wave interference pattern since both are a measure of the difference between two paths.
Problems with Interferometers
Interferometers require an arm with well known physical characteristics (such as free space) to act as a reference arm. Since the reference is simulated, all characteristics of the reference path are well known and any arbitrary characteristics can be chosen for the reference path. We use free space as the reference path in the examples for simplicity and by analogy to physically referenced interferometers. The use of free space in the examples is not meant to limit the generality of the virtual reference technique and the types of virtual references that can be simulated. Free space is used as it simplifies the expressions developed in later sections. The reference arm can also be used to balance the delay in the test arm (FIG. 1). The use of physical components in the reference arm, however, adds to its cost, complexity and limits its performance.
The following discussion considers the issues for an optically based reference arm in order to illustrate these issues in a tangible way. It is by no means exhaustive and serves as an example of the many issues encountered in interferometry in general. The issues can extend to an interferometer that acts on any type of wave.
The first issue in making an interferometer are that the components and labour cost of producing the reference arm are high. Since the reference path must have known spatial and material properties a free space path is often chosen as a reference. The components required to make a variable free space path (optical delay line) for the reference arm include high precision translation stages, waveguide to free space collimators and high precision adjustable mirrors/reflectors/retroreflectors. The components required can cost in the tens of thousands of dollars. The component costs, however, comprise only a fraction of the system cost. The fabrication and alignment of the free space path is extremely complex and difficult since it must achieve waveguide-to-free-space and free-space-to-waveguide coupling over the entire variable delay length. This becomes more difficult as the required variabilty increases and is especially difficult for folded free space paths that use folded optics to compress the footprint of an optical delay. Since commercial instruments would generally require a variability in the reference length of greater than 1 meter and given that optical waveguides generally have core sizes in the micron range this can mean a distance-to-target vs. target-size ratio that can be as high as a million to one, which is simply not feasible.
The next issues in creating an interferometer come from the performance limitations inherent in the free space reference arm.
The first performance issue is related to the dynamic range of the interferometer which is the variability of the reference arm. The dynamic range is limited by several factors. The first is that the translation stage itself limits the variation in the delay that can be achieved since it has a finite maximum and minimum length. The second limit on the dynamic range is that the collimation lens(es) used to couple light between fiber and free space have an optimum operating distance and the coupling loss increases as the delay is varied from this optimum. This increase in loss ultimately limits the variability of the delay depending on the amount of loss that the system can afford. The third limit on the dynamic range of the interferometer is determined by the precision with which the optics can be aligned. A longer more dynamic delay requires higher precision and tighter controls during the alignment process. Ultimately, however, a limit will be reached where higher precision is not possible and the delay cannot be made longer or more variable. The fourth limit on the dynamic range of the interferometer is determined by the how well the translation stage can maintain straight and flat axial path as it moves from one position to the next. This is important in determining whether or not the alignment can be maintained as a function of position. Ultimately, the dynamic range will be set by the combination of all four of these factors. It is no wonder why the commercially available optical delay lines do not have a very wide dynamic range.
The second performance issue is related to the step size of the translation stage, which is the smallest distance that the delay line can move. This smallest step size is a factor that determines the smallest separation between data points taken as the length of the reference arm is varied. If the system includes folded paths then the consecutive points will be separated further by twice the number of folds. As a result if the frequency of the radiation used to generate the interference is high (such as at optical frequencies) then the incremental step size must be at least sub micron so that data points taken at each position will be close enough together to produce a useful plot. This makes the requirements on the optical stage more stringent and it therefore means that a more costly stage must be used required.
The third performance limitation is due to nonidealities introduced by the reference arm. For example optics (mirrors, lenses etc) are generally introduced into the free space path and sometimes air is present in the path. What this means is that the ‘free space path’ is not really free space anymore because the optics and air change the reference. For example the lens introduces a small amount of dispersion in the reference path which must be known or calibrated out of the system and thus introduces error. The air flow in the free space path introduces instability related to flow and temperature fluctuations. The only way to control this issue is to use air flow and temperature stabilization techniques (i.e. placing the reference path in vacuum). This makes the system very costly.
The fourth performance issue is the time required to perform an experiment. For each data point in the experiment the system must first move the reference arm to a specified location (time) and then sample the interference pattern at that location (more time). This means that for every measurement point the system produces we must wait for it to both move and perform a scan. This severely limits the speed with which we can produce a plot with multiple data points.
The issues experienced when using a physical reference path lead to increases in the cost of interferometric systems and drastically reduce their performance. Thus, what is required is means for eliminating the physical reference path to decrease cost and increase performance.